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Properties

Label 1-571-571.131-r0-0-0
Degree $1$
Conductor $571$
Sign $0.999 + 0.0205i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.986 − 0.164i)2-s + (−0.879 − 0.475i)3-s + (0.945 + 0.324i)4-s + (−0.401 + 0.915i)5-s + (0.789 + 0.614i)6-s + (−0.0825 − 0.996i)7-s + (−0.879 − 0.475i)8-s + (0.546 + 0.837i)9-s + (0.546 − 0.837i)10-s + (0.546 + 0.837i)11-s + (−0.677 − 0.735i)12-s + (0.245 − 0.969i)13-s + (−0.0825 + 0.996i)14-s + (0.789 − 0.614i)15-s + (0.789 + 0.614i)16-s + (0.789 + 0.614i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (−0.879 − 0.475i)3-s + (0.945 + 0.324i)4-s + (−0.401 + 0.915i)5-s + (0.789 + 0.614i)6-s + (−0.0825 − 0.996i)7-s + (−0.879 − 0.475i)8-s + (0.546 + 0.837i)9-s + (0.546 − 0.837i)10-s + (0.546 + 0.837i)11-s + (−0.677 − 0.735i)12-s + (0.245 − 0.969i)13-s + (−0.0825 + 0.996i)14-s + (0.789 − 0.614i)15-s + (0.789 + 0.614i)16-s + (0.789 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.999 + 0.0205i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.999 + 0.0205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5561016286 + 0.005707929421i\)
\(L(\frac12)\) \(\approx\) \(0.5561016286 + 0.005707929421i\)
\(L(1)\) \(\approx\) \(0.5252320416 - 0.04471791576i\)
\(L(1)\) \(\approx\) \(0.5252320416 - 0.04471791576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.986 - 0.164i)T \)
3 \( 1 + (-0.879 - 0.475i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
7 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (0.546 + 0.837i)T \)
13 \( 1 + (0.245 - 0.969i)T \)
17 \( 1 + (0.789 + 0.614i)T \)
19 \( 1 + (-0.879 - 0.475i)T \)
23 \( 1 + (-0.879 + 0.475i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.245 + 0.969i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.546 - 0.837i)T \)
47 \( 1 + (0.789 + 0.614i)T \)
53 \( 1 + (-0.986 + 0.164i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (-0.986 + 0.164i)T \)
71 \( 1 + T \)
73 \( 1 + (0.245 + 0.969i)T \)
79 \( 1 + (0.945 - 0.324i)T \)
83 \( 1 + (0.789 + 0.614i)T \)
89 \( 1 + (0.789 + 0.614i)T \)
97 \( 1 + (0.945 + 0.324i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.52873844644207317457939598289, −22.37046098581564956810254557863, −21.22581398256576461531244236365, −21.02738359942824511797490947726, −19.730744410576977776269300218835, −18.89291368946809845144444344468, −18.29485370378093656005784657057, −17.18028623437351746941100601371, −16.4105878348733532463481388933, −16.18190610844325275718590083268, −15.23570171237953244062445748771, −14.19615422062324631955374000339, −12.45355732419089669091144346057, −11.96718755989746395569219817833, −11.30849799949249117546175517505, −10.26146929276230487418547569826, −9.128453176095200795433137927286, −8.86691784704432863863215800003, −7.70212054134820162159797659417, −6.31909900194962573387925976055, −5.85726569660292183238591794463, −4.72912265400149151114592900944, −3.528943999191505884129266153428, −1.89182984786401311440851798714, −0.680887527539972932655503100374, 0.80720645052045102733573231393, 1.96146092107991536816303810531, 3.29637467980688011246538286804, 4.40200539018564037015117661117, 6.13829914844277691004182381397, 6.63759759154961691453950408955, 7.6303058855814924956770407485, 8.05056982613533800968424016320, 9.86615643715269038106212957131, 10.36277868126911442002516004576, 11.048203658966824103409613884804, 11.94479711178175903093061218425, 12.68034226324781116085653345497, 13.87546875803336534830531202112, 15.12843207198862677219437148199, 15.79407472519191994357228732446, 17.00771426151490525228358613569, 17.39532022155980800533685832492, 18.10352238976944313235871041758, 19.09129229427208584393758640749, 19.60006480149241626947722803006, 20.47846364560005923745736112539, 21.65936654499601033628384729476, 22.52329835316437519485852979537, 23.33821780312617651664780882906

Graph of the $Z$-function along the critical line